Bruhat-Tits theory from Berkovich's point of view. II. Satake compactifications

TL;DR

This paper explores Berkovich's non-Archimedean analytic geometry to understand compactifications of Bruhat-Tits buildings, linking them to representation theory and providing new embedding constructions.

Contribution

It establishes that Berkovich compactifications coincide with previous constructions and introduces a natural embedding method using Berkovich geometry.

Findings

Berkovich compactifications match earlier glued constructions.

Compactifications can be recovered via linear representations.

New equivariant embedding construction based on Berkovich geometry.

Abstract

In our previous paper "Bruhat-Tits theory from Berkovich's point of view. I ? Realizations and compactifications of buildings", we investigated realizations of the Bruhat-Tits building B(G,k) of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry, and we studied in detail the compactifications of the building which arise from this point of view. In this paper, we give a representation theoretic flavor to these compactifications, following Satake's original constructions for Riemannian symmetric spaces. We first prove that Berkovich compactifications of a building coincide with the compactifications previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding B(G,k) in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.